The interrelationship of integrable equations, differential geometry and the geometry of their associated surfaces

Transcription

1 arxiv: v [math-ph] 18 May 009 The interrelationship of integrable equations, differential geometry and the geometry of their associated surfaces Paul Bracken Department of Mathematics University of Texas Edinburg, TX USA January 009 Abstract A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it is used in several subjects which are discussed. Structure equations are introduced in terms of differential forms. Forms are shown to be very useful in relating geometry, equations and surfaces, which appear in many sections. The topics of the chapters are different and separate, but joined together by common themes and ideas. Several subjects which are not easy to access are elaborated, such as Maurer-Cartan cocycles and recent results with regard to generalizations of the Weierstrass-Enneper method for generating constant mean curvature surfaces in three and higher dimensional Euclidean spaces. MSc: 53A05, 53A10, 53C15, 53C80, 53Z05, 35Q51 bracken@panam.edu 1

2 Table of Contents. 1. Introduction Classical Theory of Surfaces in Euclidean Three-Space Surfaces on Lie Algebras, Lie Groups and Integrability Differential Forms, Moving Frames and Surfaces Introduction to Moving Frames SO(3) Lax Pair Nonlinear Partial Differential Equations Admitting SO(, 1) Lax Pairs Maurer-Cartan Cocycles and Prolongations The Generalized Weierstrass System Inducing Surfaces of Constant and Nonconstant Mean Curvature Generalized Weierstrass Representations A Physical Application Involving Nonlinear Sigma Models Non-Constant Mean Curvature Surfaces References

3 1. INTRODUCTION The origins of soliton theory go back to the early part of the nineteenth century, in particular, to the observation of John Scott Russell in 1834 of a solitary bump-shaped wave moving along a canal near Edinburgh. However, it wasn t until 1965 that this type of phenomena was rediscovered, in particular, by Kruskal and Zabusky [1,] in the context of the Fermi-Pasta-Ulam [3] problem. It was they that coined the term soliton. In 1895, two Dutch mathematicians, Korteweg and de-vries derived a nonlinear wave equation which now bears their name. It models long wave propagation in a rectangular channel and has a traveling wave solution which resembles the solitary wave observed by Russell. In fact, a pair of equations equivalent to the KdV equation appeared even earlier in a work by Boussinesq. It was not until the mid-twentieth century that the equation reappeared in work by such researchers as Kruskal and Zabusky and Gardener and Morikawa in 1960 in an analysis of the transmission of hydromagnetic waves. There continues to be ongoing interest in such nonlinear equations which arise in a diversity of systems such as the theory of solids, liquids and gases [4,5]. Self-localized nonlinear excitations are fundamental and inherent features of quasi-one-dimensional conducting polymers. In 196, Perring and Skyrme solved the sine-gordon equation numerically while using it in an elementary particle model. The results generated from this equation were found not to disperse and two solitary waves were seen to keep their original shapes and velocities despite collisions. In the pioneering work of Kruskal and Zabusky, the KdV equation was obtained as a continuum limit of an anharmonic lattice model with cubic nonlinearity. The model displays the existence of solitary waves. These waves have the remarkable property that they preserve both their amplitude and speed upon interaction. These properties are the main reason for the use of the term soliton. A soliton or solitary wave can be regarded as a solution to any number of a variety of nonlinear partial differential equation. In a more physical language, solitons have the following striking properties. Energy is localized within a small region and an elastic scattering phenomenon exists in the interaction of two solitons. To put it another way, the shape and velocity of the wave are recovered after an interaction between such solutions. Solitons seem to behave as both particle and wave. Originally they arose in the area of fluid mechanics, and their study has extended into the areas of plasma physics, nonlinear optics and classical and quantum field theory. In large part, this is precisely due to the aforementioned properties. What is more, many branches of mathematics and physics provide important tools for the study of solitons. It will be seen here that the development of the study of solitons has resulted in reciprocal advances in many areas of mathematics as well. Moreover, there is a deep connection between many of these equations, the theory of surfaces [6] and integrable systems [7,8]. It is the intention here to explore this interrelationship between the theory of these equations and the surfaces that can be determined 3

4 by them. It will be seen that many important ideas from the area of differential geometry are applicable to the subjects studied here and give the subject a unified perspective. A generic method for the description of soliton interaction begins with a transformation which was originally introduced by Bäcklund to generate pseudospherical surfaces. Later Bianchi showed that the Bäcklund transformation admits a commutativity property, a consequence of which is a nonlinear superposition principle which is referred to as the permutability theorem. As an example, both KdV and MKdV equations reside in hierarchies which admit auto-bäcklund transformations, nonlinear superposition principles as well as multi-soliton solutions. This article is a review of recent results by the author as well as by other researchers. Let us begin with a brief overview of the contents which follow. Although there are numerous ideas and themes which run throughout the article, each chapter is separate and can be read on its own independently of the others. First, a review of surface theory from the classical point of view will be presented [6,9]. The next section is a novel and active area of interest which should appeal to those with an interest in this area. The subject of the immersion of a two-dimensional surface into a three-dimensional Euclidean space, as well as the n-dimensional generalization, has been related to the problem of investigating surfaces in Lie groups and in Lie algebras as well [10,11]. This gives an interesting correspondence between the Lax pair of an integrable equation and their integrable surfaces. Using the formulation of the immersion of a two-dimensional surface into three-dimensional Euclidean space, it will be shown that a mapping from each symmetry of integrable equations to surfaces in R 3 can be established. Next a differential forms approach to surfaces will be presented [1-14]. The problem of identifying whether a given nonlinear partial differential equation admits a linear integrable system is studied here by means of this differential geometric formalism [15]. It is shown that the fundamental equations of surface theory can be used to reproduce the compatibility conditions obtained from a linear system in matrix form corresponding to a number of different Lie algebras. In fact, the system of equations which has been obtained from the linear matrix problem is derived from a system of differential forms and in combination with the first and second fundamental forms leads to a link with surface theory in differential geometry [16]. The subject of non-linear evolution equations and Maurer-Cartan cocycles on R is introduced next. Maurer-Cartan cocycles are defined and some general theoretical information about them is provided. By using Maurer-Cartan cocycles and Cartan prolongations for individual nonlinear equations, it is shown how Bäcklund transformations can be calculated for specific equations. As an example, the Bäcklund transformation for the sine-gordon equation will be derived. Finally, the subject of constant mean curvature surfaces has been of great interest recently. From what has been discussed already, the theory of surfaces has many applications in a great number of areas of physical science. The theory of constant mean 4

5 curvature surfaces has had a great impact on many problems which have physical applications. In particular, there are many applications to such areas as two-dimensional gravity, quantum field theory, statistical physics and fluid dynamics [17,18]. An application of recent interest is the propagation of a string through space-time, in which the particle describes a surface called its world sheet. Thus, the subject of generalized Weierstrass representations, in particular, the generalization of the Weierstrass-Enneper approach due to B. Konopelchenko [19] will be discussed in detail. There exists a correspondence between this representation and the two-dimensional nonlinear sigma model. Both of these systems have been shown to be integrable, and their symmetry groups have been calculated [0]. These symmetries have lead to the calculation in closed form of many explicit solutions of the system, and the determination of their soliton surfaces, as will be seen.. CLASSICAL THEORY OF SURFACES It is perhaps useful at this point to introduce some classical results concerning surfaces, which arise out of classical differential geometry. This will give a basic review of surface theory and some preparation for what is to follow. Let r = r(u, v) denote the position vector of a generic point P on a surface Σ in R 3. Then the vectors r u and r v are tangential to Σ at P and at such points at which they are linearly independent, the quantity N = r u r v r u r v, (.1) determines the unit normal to Σ. The first and second fundamental forms of Σ are defined by I = ds I = dr dr = E du + F dudv + G dv, (.) II = ds II = dr dn = e du + fdudv + g dv. In (.), the coefficients are defined by E = r u r u, F = r u r v, G = r v r v, e = r u N u, g = r v N v, f = r u N v = r v N u. (.3) There is a classical result of Bonnet which states that {E, F, G, e, f, g} determines the surface Σ up to its position in space. The Gauss equations associated with Σ are given as r uu = Γ 1 11 r u +Γ 11 r v +en, r uv = Γ 1 1 r u +Γ 1 r v +fn, r vv = Γ 1 r u +Γ r v +gn, (.4) 5

6 while the Weingarten equations are given by N u = ff eg ef fe r H u + r H v, N v = gf fg ff ge r H u + r H v, (.5) where H = r u r v = EG F. The Γ i jk are the Christoffel symbols and since the derivatives of all the {E, F, G, e, f, g} with respect to u and v can be calculated from (.3) and (.4), the derivatives of all the Γ i jk can be calculated as well. Thus, using these derivatives, the compatibility conditions (r uu ) v = (r uv ) u and (r uv ) v = (r vv ) u applied to the linear Gauss system (.4) produces the nonlinear Mainardi- Codazzi system e v f u = eγ 1 1+f(Γ 1 Γ 1 11) gγ 11, f v g u = eγ 1 +f(γ Γ 1 1) gγ 1. (.6) The Theorema egregium of Gauss provides an expression for the Gaussian or total curvature eg f K = EG F, (.7) or in terms of E, F, and G alone in Liouville s representation K = 1 H [(H E Γ 11 ) v ( H E Γ 1 ) u]. (.8) If the total curvature of Σ is negative, that is, if Σ is a hyperbolic surface, then the asymptotic lines on Σ may be taken as parametric curves. Then e = g = 0 and the Mainardi-Codazzi equations reduce to Moreover, we have ( f H ) u + Γ f 1 H = 0, ( f H ) v + Γ 1 f 1 H K = f H = 1 ρ, = 0. (.9) Γ 1 1 = GE v FG u, Γ H 1 = EG u FE v. (.10) H The angle between the parametric lines is such that cosω = and since E, G > 0, we may take without loss of generality F, sin ω = H, (.11) EG EG E = ρ a, G = ρ a, f = abρ sin ω. (.1) 6

7 Then the Christoffel symbols are given by Γ 1 1 = ρ va + ρa v cosω(ρ u b + ρb u ) ρa sin, Γ 1 ω = ρ ub + ρb u cosω(ρ v a + ρa v ) ρb sin. ω (.13) Substituting (.13) into the pair (.9), there results ρa v +ρ v a cos ω(ρ u b+ρb u ) ρ v a sin ω = 0, ρb u +ρ u b cosω(ρ v a+ρa v ) ρ u b sin ω = 0. (.14) Solving the linear system in (.14) for a v and b u, we obtain a v + a ρ v ρ 1 bρ u ρ cos ω = 0, b u + b ρ u ρ 1 aρ v cosω = 0. (.15) ρ The representation for the total curvature is ω uv + 1 (ρ u ρ b a sin ω) u + 1 (ρ v a ρ b sin ω) v ab sin ω = 0. (.16) For the particular case in which K = 1/ρ < 0 is a constant, Σ is referred to as a pseudospherical surface. Then (.15) implies that a = a(u), b = b(v), and if Σ is now parametrized by arc length along asymptotic lines, then ds I = du + cosω dudv + dv, ds II = sin ω dudv. (.17) ρ Equation (.16) then reduces to the sine-gordon equation ω uv = 1 sin ω. (.18) ρ Thus, there is a clear indication of a relationship between surfaces and an integrable equation. 3. SURFACES ON LIE ALGEBRAS, LIE GROUPS AND INTEGRABILITY There have been some interesting developments recently related to the problem of the immersion of a -dimensional surface into a 3-dimensional Euclidean space, as well as the n-dimensional generalization [1,,3]. These will be reviewed here. This subject has been shown to be related to the problem of studying surfaces in Lie groups and Lie algebras [4]. It has been found useful for investigating integrable surfaces, or surfaces which are described by integrable equations. Starting from a suitable Lax pair, which implies a suitable integrable equation, it is possible to construct explicitly large classes of integrable surfaces. 7

8 Let F = (F 1, F, F 3 ) : π R 3 be an immersion of a domain π R into 3- dimensional Euclidean space. For (u, v) π, the Euclidean metric induces some metric with coefficients g ij (u, v) on the surface. These functions and d ij (u, v), which define the second fundamental form, satisfy a system of three nonlinear equations known as the Gauss-Codazzi equations, which are the compatibility condition of the Gauss- Weingarten system. There exist two geometrical characteristics on such a surface known as the Gauss curvature K and the mean curvature H. Some results will be given in other sections which correspond to constant K and constant H. A surface will be called integrable if and only if its Gauss-Codazzi equations are integrable. Integrable equations also arise as the compatibility condition of a pair of linear equations, which is usually referred to as a Lax pair. Here we want to show this problem is closely related to the problem of studying surfaces in Lie groups and Lie algebras. Let G be a group and G the Lie algebra of G. Assume there exists an invariant scalar product in G. The scalar product will not be degenerate so there exists an orthonormal basis {e i } in G such that e i, e j = δ ij. To introduce a surface in G, let Φ(u, v) G for every (u, v) in some neighborhood of R. There exists a canonical map from the tangent space of G to the Lie algebra G. If Φ u and Φ v are the tangent vectors of Φ at the point (u, v), this map is defined by Φ u Φ 1 = U j e j, Φ v Φ 1 = V j e j, (3.1) where U j and V j are some functions of (u, v) and j = 1,, n. Equations (3.1) define Φ through its value in the Lie algebra. Suppose the structure constants of G satisfy [e k, e m ] = c kmj e j, (3.) with summation implied. Differentiating the first equation of (3.1) with respect to v and the second with respect to u, then upon subtracting these we have U j v e jφ+u j e j Φ v V j u e jφ V j e j Φ u = ( U j v V j u )e jφ+(u m V s e m e s V s U m e s e m )Φ = 0. (3.3) Expression (3.3) implies that ( U j v V j u )e j + U m V s c msj e j = 0. which when written just in terms of U and V, this is written U v V u + [U, V ] = 0. (3.4) 8

9 This result can be summarized next. Proposition 3.1. Let Φ(u, v) G be a differentiable function of u, v for every (u, v) in some neighborhood of R. Then Φ defined by (3.1) exists if and only if the functions u j and V j satisfy (3.4). To introduce a surface in G, let F(u, v) G for every (u, v) in a neighborhood of R. The first fundamental form of F is defined by F u, F u du + F u, F dudv + F v v, F v dv. (3.5) Let N (s) G, s = 1,, n be the elements of G defined by N (l), N (l) = 1, F u, N (l) = F v, N (l) = 0. Then the second fundamental forms of F are defined by F u, N(s) du + F u v, N(s) dudv + F v, N(s) dv, (3.6) for s = 1,, n. Surfaces in G can be related to surfaces in G by using the adjoint representation to write F u = Φ 1 a j e j Φ, F v = Φ 1 b j e j Φ, (3.7) where a j and b j are some functions of (u, v). Differentiating the first expression in (3.7) with respect to v and using the fact that (Φ 1 ) τ = Φ 1 Φ τ Φ 1, then modulo (3.7), we obtain F u v = Φ 1 V s e s a j e j Φ+Φ 1 a j v e jφ+φ 1 a j e j V s e s Φ = Φ 1 ( a j v e j V s a m c smj e j )Φ. In a similar way, differentiating F v with respect to u, we have (3.8) F u v = Φ 1 ( b j u e j U s b m c smj e j )Φ. (3.9) Requiring that the derivatives in (3.8) and (3.9) match gives the following result. Proposition 3.. Let Φ(u, v) G be a surface defined by (3.1). Let F(u, v) G be a differentiable function of u and v for every (u, v) in some neighborhood of R. Then (3.7) defines a surface F(u, v) G if and only if a j and b j satisfy a j v + a kv m c kmj = b j u + b kv m c kmj, k, m, j = 1,, n. It is often possible to calculate a j, b j and F explicitly. Theorem 3.1. Let U j (u, v) and V j (u, v), j = 1,, n be differentiable functions of u and v for every (u, v) in some neighborhood of R. Let {e j } n j=1 be an orthonormal basis in the Lie algebra G of the Lie group G. 9

10 Suppose that U j and V j depend on a parameter λ and satisfy (3.3), where c kmj, k, m, j = 1,, n are the structure constants associated with G, but λ does not appear explicitly in (3.4). Define U and V as follows (i) If A and B are defined to be U = U j e j, V = V j e j. (3.10) U A = a j e j = α 1 u + α U v + α U 3 λ + α 4 u (uu) + α 5v U + [U, M], v V B = b j e j = α 1 u + α V v + α V 3 λ + α 4u V u + α 5 (vv ) + [V, M], v (3.11) where M = m j e j and α 1,, α 5, m 1,, m n are constant scalars, then the equations F u = Φ 1 AΦ, F v = Φ 1 BΦ, (3.1) are compatible, and can be used to define a surface F(u, v) G. (ii) The solution of (3.1) where A and B are defined by (3.11) is, to within an additive constant, given by F = α 1 Φ 1 UΦ+α Φ 1 V Φ+α 3 Φ 1 Φ λ +α 4uΦ 1 UΦ+α 5 vφ 1 V Φ Φ 1 MΦ. (3.13) Proof: The equations (3.1) are compatible if and only if (3.4) holds. From the equations for F, we determine that F v u = Φ 1 v AΦ+Φ 1 A v Φ+Φ 1 A Φ v = Φ 1 V m e m AΦ+Φ 1 A v Φ+Φ 1 AV m e m Φ = Φ 1 ( A [V, A])Φ. (3.14) v Similarly, F u v = Φ 1 ( B + [B, U])Φ. (3.15) u Upon equating the derivatives in (3.14) and (3.15) and moving all terms to the same side, it follows that A v B + [A, V ] + [U, B] = 0. (3.16) u Suppose A and B are defined by (3.11) and U and V satisfy (3.4), then by direct calculation, we have A v B u = α 1 u ( U v V u )+α v ( U v V u )+α 3( U v λ V u λ )+α 4( v (uu) u V u ) 10

11 +α 5 v (v U v (vv )) + [ U, M] [ V u v u, M] = α 1 u [U, V ] α v [U, V ] α 3 λ [U, V ] α 4 u (u[m, V ]) α 5 (v[u, V ]) (3.17) v +[ U v V u, M]. Using A and B given in (3.11), let us work out the terms of [A, V ] + [U, B] according to each coefficient α j one at a time, α 1 [ U u, V ] + α 1[U, V u ] = α 1 [U, V ], u α [ U v, V ] + α [U, V v ] = α [U, V ], v α 3 [ U λ, V ] + α 3[U, V λ ] = α 3 [U, V ], (3.18) λ α 4 [ u (uu), V ] + α 4[U, u V u ] = α 4 (u[u, V ]), u α 5 [v U v, V ] + α 5[U, v (vv )] = α 5 (v[u, V ]). v Finally, using Jacobi s identity, we can write [[U, M], V ]+[U, [V, M]] = [V, [M, U]]+[U, [V, M]] = [[U, V ], M] = [ U v V u, M]. Substituting all of these results for the brackets (3.18) as well as for A v B u from (3.17) into the left-hand side of (3.16), it can be seen that (3.16) is satisfied identically. (ii) To prove that F given by (3.13) satisfies (3.1), differentiate F with respect to u to obtain F u = α 1Φ 1 UUΦ+α 1 Φ 1 U u Φ+α 1Φ 1 UUΦ α Φ 1 UV Φ+α Φ 1 V u Φ+α Φ 1 V UΦ α 3 Φ 1 U Φ λ +α 3Φ 1 Φ u λ +α 4Φ 1 UΦ α 4 uφ 1 UUΦ+α 4 uφ 1 U u Φ+α 4uΦ 1 UUΦ α 5 vφ 1 UV Φ + α 5 vφ 1 V u Φ + α 5vΦ 1 V UΦ + Φ 1 UMΦ Φ 1 MUΦ = Φ 1 U {α 1 u +α ( V u [U, V ]) α U 3 λ +α 4 u (uu)+α 5(v V v[u, V ])+[U, M]}Φ. u 11

12 Using (3.4), this simplifies to the form F u = U Φ 1 {α 1 u + α V v α U 3 λ + α 4 u (uu) + α 5v U v + [U, M]}Φ = Φ 1 AΦ, as required. Similarly, the derivative of F with respect to v is calculated in the same way, and the second equation of (3.1) then results. Using a variation of parameter, it follows that this F is unique to within a constant matrix. This is really a consequence of the fact that (3.16) is the variational equation of (3.4). In fact, if U and V are replaced by U + ǫa and V + ǫb, then the O(ǫ) term of (3.4) yields (3.16). This means that every symmetry of (3.4) implies a solution of (3.16). It will be useful and instructive to write down the previous Theorem for the case in which the group G is SU(). In this case, e j = iσ j, for j = 1,, 3 where σ j are the Pauli matrices given by σ 1 = ( ), σ = ( 0 i i 0 ) ( 1 0, σ 3 = 0 1 ), (3.19) and the structure constants are given by c ijk = ǫ ijk, where ǫ ijk, i, j, k = 1,, 3 is the usual antisymmetric tensor. To the vector F = (F 1, F, F 3 ) T R 3, we associate the matrix F = F j e j su(), which we write ( ) if3 F F = if j e j = if 1. (3.0) F if 1 if 3 The problem of immersing the -dimensional surface x j = F j (u, v), j = 1,, 3 into 3-dimensional space becomes the problem of studying the relationship between the 3- dimensional sphere Φ(u, v) SU() and the two-dimensional surface F(u, v) su(). Thus taking e j = iσ j and c ijk = ǫ ijk in Theorem 3.1, this theorem can be restated for the case of SU(). Theorem 3.. Let U(u, v) and V (u, v) su() be differentiable functions of u and v for every (u, v) in some neighborhood of R. Assume that the functions U and V satisfy equation (3.4). Then the equations Φ u = UΦ, Φ v = V Φ, (3.1) define a -dimensional surface Φ(u, v) SU(). Let A(u, v) and B(u, v) su() be real, differentiable functions of u and v for every (u, v) in some neighborhood of R. In addition to this, assume that these functions satisfy (3.16). Then equations (3.1) together with F = if j σ j define a -dimensional 1

13 surface x j = F j (u, v) R, j = 1,, 3 in a 3-dimensional Euclidean space. The first and second fundamental forms of this surface are and A, A du + A, B dudv + B, B dv, (3.) A u + [A, U], C du + A + [A, V ], C dudv + B v v + [B, V ], C dv, (3.3) respectively. In (3.) and (3.3), we have A, B = 1 [A, B] tr (A, B), C = [A, B], A = A, A. (3.4) Let us consider the following example. In Theorem 3., let us put A = iaσ 1, B = i(b 1 σ 1 + b σ ), U = i U jσ j and V = i V jσ j into (3.16). After using the commutation relations, we obtain ( a v + b 1 u +b U 3 )σ 1 +( b u +av 3 b 1 U 3 )σ +( av +b 1 U b U 1 )σ 3 = 0. (3.5) For this to hold, the coefficients of σ j must vanish giving the system of equations a v b 1 u b U 3 = 0, The first and third equations of (3.6) imply that b u +av 3 b 1 U 3 = 0, av b 1 U +b U 1 = 0. (3.6) U 3 = 1 b ( a v b 1 u ), V = 1 a [b 1U b U 1 ]. Putting these results in the second equation of (3.6) gives V 3 = 1 a [b 1 ab v b b 1 1 u b b u ]. Let Φ SU(), then the equations for F are obtained by substituting A and B. They take the form F u = Φ 1 aσ 1 Φ, Then F defines a -dimensional surface in R 3. Moreover, F v = iφ 1 (b 1 σ 1 + b σ )Φ. (3.7) A, A = a, A, B = ab 1, B, B = b 1 + b, 13

14 give the coefficients of the first fundamental form (3.), which can be written I = a du + ab 1 dudv + (b 1 + b ) dv. (3.8) With A = iaσ 1 and B = i(b 1 σ 1 +b σ ). we calculate [A, B] = [aσ 1, b 1 σ 1 +b σ ] = ab σ 3 and C = σ 3. Therefore, [A, U] = 1 au j[σ 1, σ j ] = au σ 3 and A u + [A, U], C = i a u σ 1 au σ 3, σ 3 = au. Similarly, it follows that [A, V ] = a(σ 3 V σ V 3 ), and we must have A v + [A, V ], C = av, B v + [B, V ], C = b 1V b V 1. Putting these together in (3.3), the second fundamental form is given by II = au du + av dudv + (b 1 V b V 1 ) dv. (3.9) In terms of matrices, these fundamental forms are given by ( ) ( a ab I = 1 au av ab 1 b 1 + b, II = av b 1 V b V 1 ). (3.30) The Gauss and mean curvature are defined by K = det(ii I 1 ) = ( U 1 a ) + U a (b 1U 1 av 1 ), H = tr(ii I 1 ) = U ab a b 1U 1 av 1. ab (3.31) It can be shown the surface F is unique up to position in space. Given the fundamental forms (3.30), U, V and V 1 can be solved for. Since these functions satisfy the Gauss- Codazzi equations (3.4), Φ SU() can be defined by (3.1) to within three constants. Equations (3.1) imply F su() within three additional constants.these six arbitrary constants correspond to arbitrary motions of the surface in R 3. Indeed, the transformations ˆF = fff 1 + Ã, ˆN = fnf 1, ˆΦ = Φf, f SU(), à su() leave (3.1) and the fundamental forms invariant. The constants of à introduce a translation while the constants of f introduce a rotation, thus six arbitrary constants appear. Let us summarize this collection of results in Theorem 3.3. Theorem 3.3. Let U 1, U, V 1, a, b 1 and b such that a 0, b 0 be real differentiable functions of u and v for every (u, v) in some neighborhood of R. Assume that these functions satisfy the Gauss-Codazzi equations (3.4), where U 3, V and V 3 are defined by U 3 = 1 ( a b v b 1 u ), V = 1 a (b 1U b U 1 ), V 3 = 1 a (b 1 ab v b b 1 1 u b b u ). 14 (3.3)

15 Let Φ SU() be defined by (3.1). Then the equations F u = iφ 1 aσ 1 Φ, F v = iφ 1 (b 1 σ 1 + b σ )Φ, (3.33) where σ j are the Pauli matrices (3.19), define a -dimensional surface x j = F j (u, v), j = 1,,, (3.34) in R 3. Its first and second fundamental forms are given in (3.8) and (3.9). The Gauss and mean curvatures are given in (3.31). This surface is unique to within position in space. To close this section, a final result and an application along these lines is presented below and gives an explicit construction of functions A and B as well as the immersion function F based on the symmetries of (3.4) and (3.1) [5]. Theorem 3.4. Suppose that U(u, v) and V (u, v) can be parametrized in terms of λ and a scalar function θ(u, v) in such a way that (3.4) is equivalent to a single partial differential equation for θ(u, v) independent of Λ. This equation, which by definition is called an integrable partial differential equation, possesses the Lax pair defined by (3.1). Define the su() valued functions A(u, v, λ) and B(u, v, λ) by A = α U λ + M u + [M, U] + U φ, (3.35) B = α V λ + M v + [M, V ] + V φ, (3.36) where α(λ) is an arbitrary scalar function of λ. Also, M(u, v; λ) is an su() valued arbitrary function of u, v and λ and the scalar φ is a symmetry of the partial differential equation satisfied by the function θ(u, v). The prime denotes Fréchet differentiation. Then there exists a surface with immersion F(u, v; λ) defined in terms of A, B, Φ by (3.35) and (3.36). Furthermore, F to within an additive constant, is given by F = Φ 1 (α Φ λ + MΦ + Φ φ). (3.37) Proof: This is similar to Theorem 3., so we just verify (3.16), A v = α λ (V u [U, V ]) + M v u + [ M v, U] + [M, V u [U, V ]] + (U φ) v, B u = α λ V u + M u v + [M u, V ] + [M, V u ] + (V φ) u. 15

16 Then substituting these into (3.16) and simplifying, we find that A v B u +[A, V ]+[U, B] = [M, [U, V ]]+[M, U], V ]+[U, [M, V ]]+(U φ) v (V φ) u +[U φ, V ]+[U, V φ]. Using Jacobi s identity, the first three terms combine to give zero, and the last terms are the Fréchet derivative of (3.4), so (3.16) holds. Let us apply Theorem 3.4 to the case of the sine-gordon equation which is given by ϑ uv = sin ϑ. (3.38) In (3.38), ϑ(u, v) is a real, scalar function and time is denoted by v. Define U(u, v, λ) and V (u, v, λ) in terms of the Pauli matrices as U = i ( ϑ uσ 1 + λσ 3 ), Let ϕ be a solution of the equation V = i λ (sin ϑσ cosϑσ 3 ). (3.39) ϕ uv = ϕ cosϑ, (3.40) so ϕ is considered to be a symmetry of (3.38), and solutions of (3.40) contain the geometrical and generalized symmetries of (3.38). For each ϕ, Theorem 3.4, with α = 0, M = 0 implies the surface constructed from A = i ϕ u σ 1, B = i λ ϕ(cosϑσ + sin ϑσ 3 ), (3.41) has the immersion function given by F = Φ 1 Φ (ϕ). Sine-Gordon equation (3.38) is an integrable equation and hence admits infinitely many symmetries, which are referred to as generalized symmetries. Let S be the surface generated by U, V, A, B defined by (3.39)-(3.41). The first and second fundamental forms, Gaussian curvature and mean curvatures of this surface are given by I = 1 4 (ϕ u du + 1 λ ϕ dv ), II = 1 (λϕ u sin ϑdu + 1 λ ϕϑ v dv ), (3.4) K = 4λ ϑ v sin ϑ ϕϕ u, H = λ(ϕ uϑ v + ϕ sin ϑ) ϕϕ u. (3.43) Theorem 3.5. Let S be a regular surface defined by (3.4) and (3.43) in terms of a generalized symmetry of sine-gordon equation (3.38). If S is an oriented, compact and connected surface, then it is homeomorphic to a sphere. 16

17 Proof: All compact, connected surfaces with the same Euler-Poincaré characteristic are homeomorphic. For compact surfaces, the Euler-Poincaré characteristic χ is given by χ = 1 det(g)k dudv. (3.44) π Ω From (3.43), the integrand can be worked out to be gk = λϑv sin ϑ = λ(cosϑ) v. (3.45) Hence, χ is independent of the deformations ϕ, and putting (3.45) into (3.44), we obtain χ = λ (cosϑ) v dudv. (3.46) π Ω This implies that χ has the same value for all generalized symmetries and hence for all sine-gordon deformed surfaces. It suffices to take a simple case to calculate χ. With ϕ = ϑ v, this is a sphere with χ =. Hence, all deformed surfaces have the Euler- Poincaré characteristic χ =. 4. DIFFERENTIAL FORMS, MOVING FRAMES AND SURFACES 4.1. Introduction to Moving Frames. The use of moving frames and exterior differentiation together has become a powerful tool in differential geometry [16]. Suppose f : M R N is an embedding of an m-dimensional oriented smooth submanifold in R N. The range of values for the indices is 1 i, j, k, l m, m + 1 A, B, C, D N, 1 α, β, γ, δ N. Attach an orthogonal frame (p; e 1,, e N ) to every point in M such that e i is a tangent vector of M at p, e A is a normal vector of M at p and (e 1,, e m ) and (e 1,, e N ) have the same orientation as a fixed frame (0, δ 1,, δ N ) in R N. Suppose there is a frame field on an open neighborhood U of M, which depends continuously and smoothly on the local coordinates of U. Then we usually call such a local orthogonal frame a Darboux frame on the submanifold M. There always exists a Darboux frame in a sufficiently small neighborhood of every point in M, and the following transformations apply m e i = N a ij e j, e A = j=1 B=m+1 a AB e B, (4.1) where a ij, a AB are smooth functions on U such that (a ij ) SO(m, R), (a AB ) SO(N m; R). 17

18 Denote by ω α, ω αβ the differential 1-forms obtained by pulling the relative components of moving frames in R N back to U by f. Obviously, these 1-forms on U still satisfy the structure equations dω α = N ω β ω βα, dω αβ = γ=1 N ω αγ ω γβ. (4.) γ=1 Since the origin p of the Darboux frame is in M, and e i is a tangent vector of M at p, we have m dp = ω i e i, ω A = 0, (4.3) and the ω i, 1 i m are linearly independent everywhere. Suppose i=1 I = dp dp = m (ω i ), da = ω 1 ω m. (4.4) i=1 These quantities are independent of the transformation of Darboux frame, so they are defined on the whole manifold M. They are referred to as the first fundamental form and the area element of M. With I as the Riemannian metric, the manifold M becomes a Riemannian manifold, so M has a Riemannian metric induced from R N. The equations of motion for a Darboux frame can be written de i = m N ω ij e j + ω ia e A, de B = j=1 A=m+1 m N ω Bj e j + ω BA e A, (4.5) j=1 A=m+1 where ω α, ω αβ = ω βα are the relative components which satisfy the structure equations dω i = m ω j ω ji, 0 = j=1 m ω j ω ja, (4.6) j=1 m N dω ij = ω ik ω kj + ω ia ω Aj, k=1 A=m+1 m N dω ib = ω ik ω kb + ω ia ω AB, (4.7) k=1 A=m+1 m N dω AB = ω Ak ω kb + ω AC ω CB. k=1 C=m+1 18

19 By the Fundamental Theorem of Riemannian Geometry, the first formula of (4.6) and the skew-symmetry ω ij + ω ji = 0 together imply that ω ij is the Levi-Civita connection on the Riemannian manifold M, De j = m ω ij e j. (4.8) j=1 By the first formula in (4.5), we have that De i is the orthogonal projection of de i on a tangent plane of M. By Cartan s lemma [16], it follows from the second equation of (4.6) that ω ja = m h Aji ω i, h Aji = h Aij. (4.9) i=1 Let us put II = i,a ω i ω ia e A = N A=m+1 ( m i,j=1 h Aij ω i ω j )e A. (4.10) Then II is independent of transformation of Darboux frame. It is a differential -form defined on the whole manifold M, and taking values on the space of normal vectors to M. It is called the second fundamental form of the submanifold M. The curvature form of the Levi-Civita connection on M is Ω ij = dω ij m ω ik ω kj = 1 k=1 m k,l=1 R ijkl ω k ω l, (4.11) where R ijkl is the curvature tensor. From the first formula in (4.7), we obtain R ijkl = N A=m+1 (h Ail h Ajk h Aik h Ajl ). (4.1) This is the Gauss equation for the submanifold M. The last two formulas in (4.7) are the Codazzi equations from the theory of surfaces. For hypersurfaces in a Euclidean space, the above formulas can be greatly simplified. The reason for the preceding introduction is to address the following objectives. Let us show how the structure equations for surfaces in R 3 can be used to generate integrable equations under a suitable choice of the differential forms. This can be used to make a connection between these equations and the theory of surfaces. This is on account of the fundamental theorem for hypersurfaces in R m+1. 19

20 Proposition 4.1. Suppose there exist two differential -forms I = m m (ω i ), II = h ij ω i ω j, i=1 i,j=1 where the ω i (1 i m) are linearly independent differential 1-forms depending on m variables and h ij = h ji are functions of these m variables. Then a necessary and sufficient condition for a hypersurface to exist in R m+1 with I and II as its first and second fundamental forms is: I and II satisfy the Gauss-Codazzi equations (4.1) and (4.7). Moreover, any two such hypersurfaces in R m+1 are related by a rigid motion. Now let us show how the structure equations for surfaces in R 3 can be used to generate integrable equations by choosing the differential forms appropriately. This will allow us to make a connection between integrable equations and the theory of surfaces by means of Proposition 4.1. Moreover, it will be shown that these same partial differential equations will result from the integrability condition of a particular linear system of equations [6]. Thus, the idea of a Lax pair has a geometrical connotation as well [7]. It will be seen that very many integrable equations which are of interest in theoretical physics can be generated in this way [8]. It has also been shown that a system of differential forms can reproduce the complete set of differential equations generated by an SO(m) matrix Lax pair [9]. Here it will be of interest to study how the fundamental equations of surface theory can be used to reproduce the compatibility conditions obtained from a linear system in matrix form. The approach will be from the geometrical point of view using the structure equations and particular choices for the differential forms which appear in them. Concurrently, the linear matrix problem will be worked out alongside so the equations obtained each way can be compared. The coefficient matrices for the linear systems of interest will be based on the Lie algebras so(3) and so(, 1), which are isomorphic to the Lie algebras su() and sl(, R). It will be seen that a nonlinear partial differential equation which admits an SO(3) or SO(, 1) Lax pair must be the Gauss equation of the unit sphere in Euclidean space R 3 or Minkowski space. 4.. SO(3) Lax Pair. Let us consider the so(3) algebra first. The general form for a differential equation in terms of two independent variables and a single unknown function ϕ can be given in the form G(ϕ, ϕ x, ϕ t, ϕ xx, ϕ xt, ϕ tt, ) = 0. (4.13) In (4.13), ϕ = ϕ(x, t) and ϕ α, ϕ αβ, with α, β {t, x} are the partial derivatives of ϕ with respect to x and t. For the case of an SO(3) Lax pair, it is required that there exist 0

21 two three by three antisymmetric matrices which can be expressed in the form 0 u 1 u 13 0 v 1 v 13 U = u 1 0 u 3, V = v 1 0 v 3. (4.14) u 13 u 3 0 v 13 v 3 0 such that the two linear systems Φ t = UΦ, Φ x = V Φ, (4.15) are completely integrable when ϕ satisfies (4.13). It is said that (4.13) is a partial differential equation admitting an SO(3) Lax pair (4.14). The elements u ij and v ij which appear in (4.14) will depend on ϕ and its derivatives up to a certain order. The function Φ which appears in (4.15) can be thought of as a function in R 3 or SO(3). In fact, all possible partial differential equations of the form (4.13) which do admit such Lax pairs can be determined. The integrability condition for (4.15) in terms of U and V is given as U x V t + [U, V ] = 0. (4.16) Theorem 4.1. With respect to the components of the matrices U and V defined by the matrices in (4.14), the independent component equations of (4.16) take the form u 1,x v 1,t + u 3 v 13 u 13 v 3 = 0, u 13,x v 13,t + u 1 v 3 u 3 v 1 = 0, u 3,x v 3,t + u 13 v 1 u 1 v 13 = 0. (4.17) Equations (4.16) follow by using (4.14) in (4.16) and working out all the operations to obtain the independent components of the final matrix in (4.16). It may now be asked to what extent can the equations given in (4.17) be obtained from the structure equations given earlier which govern the Darboux frame for a manifold or immersed surface M R 3. The method can be specialized to the case of a surface in R 3. Let {p; e 1, e, e 3 } be a Darboux frame with origin p in M. A set of differential one-forms must be written down which depend on the functions {u ij } and {v ij }. First, the one-forms ω i are defined to be ω 1 = u 1 dt + v 1 dx, ω = u 13 dt + v 13 dx, ω 3 = 0. (4.18) The forms which specify the connection are written as ω 1 = u 3 dt + v 3 dx, ω 13 = u 1 dt + v 1 dx, ω 3 = u 13 dt + v 13 dx. (4.19) The forms given in (4.19) satisfy ω ij + ω ji = 0, hence the connection in (4.19) is Riemannian. Therefore, it follows that dp = ω 1 e 1 + ω e = (u 1 dt + v 1 dx)e 1 + (u 13 dt + v 13 dx)e. (4.0) 1

22 The frame vectors {e i } must satisfy the equations de i = Theorem 4.. The structure equations 3 ω ij e j. (4.1) j=1 dω 1 = ω ω 1, dω = ω 1 ω 1, dω 1 = ω 1 ω 3, (4.) and the differential forms given in (4.18) and (4.19) imply the system of equations (4.17). Proof: From (4.18) and (4.19), it follows that dω 1 = (u 1,x v 1,t ) dx dt and ω ω 1 = (u 13 v 3 v 13 u 3 ) dx dt, dω = (u 13,x v 13,t ) dx dt and ω 1 ω 1 = (u 3 v 1 u 1 v 3 ) dx dt, and finally, dω 1 = (u 3,x v 3,t ) dx dt, ω 13 ω 3 = (u 1 v 13 u 13 v 1 ) dx dt. Substituting these results into (4.), it is found that system (4.17) results. In fact, it can be seen that the two remaining structure equations dω 13 = ω 1 ω 3 and dω 3 = ω 1 ω 13 simply reproduce two of the equations already given in (4.17). Since ω 1 ω 13 = 0 and ω ω 3 = 0, automatically it follows that ω 1 ω 13 +ω ω 3 = 0. Using these results for ω i and ω ij, the fundamental forms can be written down in terms of the u ij and v ij as follows I = ω 1 + ω = (u 1 + u 13) dt + (u 1 v 1 + u 13 v 13 ) dtdx + (v 1 + v 13) dx, II = h 11 ω 1 + h 1ω 1 ω + h ω = ω 1ω 13 + ω ω 3 = I, (4.3) III = ω 13 + ω 3 = I. Now ω 13 = h 11 ω 1 + h 1 ω and ω 3 = h 1 ω 1 + h ω, and since ω 13 = ω 1 and ω 3 = ω, the components h ij of II must be h 11 = h = 1, h 1 = h 1 = 0. In this case, the two expressions for II in (4.3) exactly coincide. Using this information about h ij and the definition of principle curvature, it follows that κ 1 = κ = 1. Therefore, every point of an associated surface is an umbilical point of M. If M is a connected surface on which every point is an umbilical point, then M must be a sphere or a plane. It follows that the mean curvature and the Gaussian curvature have the values H = 1 (h 11 + h ) = 1 (κ 1 + κ ) = 1, K = h 11 h h 1 = κ 1 κ = 1. (4.4) The Gauss equation for the sphere can be obtained from (4.17). Solving the first two equations for u 3 and v 3 in (4.17), we obtain u 3 = v 3 = 1 u 1 v 13 u 13 v 1 [(v 1,t u 1,x )u 1 + (v 13,t u 13,x )u 13 ], 1 u 1 v 13 u 13 v 1 [(v 13,t u 13,x )v 13 + (v 1,t u 1,x )v 1 ]. (4.5)

23 Substituting u 3 and v 3 from (4.5) into the third equation of (4.17) gives the following second order partial differential equation ( (v 1,t u 1,x )u 1 u 1 v 13 u 13 v 1 + (v 13,t u 13,x )u 13 u 1 v 13 u 13 v 1 ) x ( (v 13,t u 13,x )v 13 u 1 v 13 u 13 v 1 + (v 1,t u 1,x )v 1 u 1 v 13 u 13 v 1 ) t +u 13 v 1 u 1 v 13 = 0. (4.6) This is an equation that is of the form (4.13). Equation (4.6) is the Gauss equation for the sphere S. Therefore, the nonlinear partial differential equation (4.6) admits an SO(3) Lax pair corresponding to an equation of the type (4.13). It is convenient to refer to a partial differential equation Q as a subequation of another equation G(ϕ, ϕ t, ϕ x, ) = 0 if every solution of Q = 0 also satisfies G = 0. Clearly, if Q = 0 admits a Lax pair, then Q = 0 must be a subequation of each equation of (4.16). Conversely, if for given u 13, u 1, v 1, v 13 with u 1 v 13 u 13 v 1 0, Q = 0 is a subequation of (4.6), then Q = 0 admits a Lax Pair in which u 3, v 3 are defined by (4.5). In this sense, all possible equations admitting SO(3) Lax pairs with u 1 v 13 u 13 v 1 0 have been determined. Defining the matrix ( ) u3 u M = 13 u 1, (4.7) v 3 v 13 v 1 then if rank(m) =, we can assume that v 13 u 1 u 13 v 1 0. When rank (M) = 1, the second row of (4.7) must be a multiple of the first row. In this case, we have v 3 = σu 3, v 13 = σu 13, v 1 = σu 1. (4.8) Substituting (4.8) into the compatibility conditions (4.17), the following conservation laws result u 3,x (σu 3 ) t = 0, u 13,x (σu 13 ) t = 0, u 1,x (σu 1 ) t = 0. (4.9) Since the integrability condition (4.16) consists of only one equation, we suppose (4.13) is the first equation here, namely u 3,x (σu 3 ) t = 0. This is the integrability condition of the system ψ t = u 3 ψ, ψ x = σu 3 ψ. (4.30) In (4.30), ψ is a real function and (4.30) is a U(1) Lax pair. These results can be summarized in the form of the following Theorem. Theorem 4.3. All nonlinear partial differential equations admitting SO(3) integrable systems can be obtained in the following ways: (i) When rank (M) =, the nonlinear equation is the Gauss equation of S R 3 or its subequation and u 1, u 13, v 1, v 13 in (4.7) are any given functions of ϕ and derivatives of ϕ up to a certain order. 3

24 (ii) When rank (M) = 1, the nonlinear equation can be chosen to be the equation of a conservation law M t + N x = 0, where N 0. If u 1, u 13, v 1 and v 13 are given functions of ϕ and derivatives of ϕ up to a certain order such that u 13 v 13 u 1 v 1 0, then Theorem 4.3 gives a straightforward way of building all nonlinear partial differential equations which admit SO(3) Lax pairs. Substituting this set of functions into (4.6), the corresponding nonlinear equation (4.13) is obtained. Some examples in which this is done will be presented now. Example 1: Let u 13 = v 1 = 0, u 1 = cos(ϕ/), v 13 = sin(ϕ/). Putting these in (4.6) gives ϕ tt ϕ xx = sin(ϕ). Example : Let u 13 = v 1 = 0, u 1 = cos(ϕ/), v 13 = sinh(ϕ/) in (4.6) gives the equation ϕ tt + ϕ xx = sinh(ϕ). Example 3: Let u 13 = v 1 = 0, u 1 = v 13 = e ϕ, then the Liouville equation is obtained ϕ tt + ϕ xx = e ϕ. Example 4: Let u 13 = v 1 = 0, u 1 = ϕ t and u 13 = ϕ, then (4.6) gives ( + ϕ ) ϕ t + ϕ xxt ϕ ϕ xϕ xt ϕ 3 = Nonlinear Partial Differential Equations Admitting SO(, 1) Lax Pairs. Consider nonlinear partial differential equations of the form (4.13) which now admit the SO(, 1) Lax pair with structure identical to (4.15), but with matrices U and V taking values in the Lie algebra so(, 1). The case in which the integrability condition for (4.15) is the Gauss equation for H R,1 will be examined. The case of S 1,1 R,1 has been examined [7]. Let us consider the case in which the relevant matrices U and V are given by U = 0 u 1 u 13 u 1 0 u 3 u 13 u 3 0, V = 0 v 1 v 13 v 1 0 v 3 v 13 v 3 0. (4.31) The compatibility condition (4.16) leads to the following Theorem. Theorem 4.4. In terms of the components of the matrices U and V defined by (4.31), the independent components of (4.16) take the form u 1,x v 1,t + u 3 v 13 u 13 v 3 = 0, u 13,x v 13,t + u 1 v 3 u 3 v 1 = 0, u 3,x v 3,t + u 1 v 13 u 13 v 1 = 0. (4.3) 4

25 These same equations can be obtained directly from the structure equations by specifying a set of differential forms. First, the one forms ω i are defined to be ω 1 = u 1 dt + v 1 dx, ω = u 13 dt + v 13 dx, ω 3 = 0. (4.33) The forms which specify the connection are given by ω 1 = u 3 dt + v 3 dx, ω 13 = u 1 dt + v 1 dx, ω 3 = u 13 dt + v 13 dx, (4.34) and satisfy ω ij = ω ji, but ω 3 = ω 3 so the connection is quasi-riemannian. Theorem 4.5. For the space H R,1, the structure equations (4.) and the differential forms (4.33) and (4.34) imply the system of equations (4.3). Proof: From (4.33) and (4.34), it follows that dω 1 = (u 1,x v 1,t ) dx dt and ω ω 1 = (u 13 v 3 u 3 v 13 ) dx dt, moreover dω = (u 13,x v 13,t ) dx dt and ω 1 ω 1 = (u 3 v 1 u 1 v 3 ) dx dt, and finally dω 1 = (u 3,x v 3,t ) dx dt, with ω 13 ω 3 = (u 13 v 1 u 1 v 13 ) dx dt. Substituting these results into (4.), the system of equations (4.3) results. The remaining two structure equations which go with (4.) simply reproduce two of the equations present in (4.3). Since both ω 1 ω 13 = 0 and ω ω 3 = 0, it follows that ω 1 ω 13 + ω ω 3 = 0. The fundamental forms can be calculated according to (4.3), and we have I = ω 1 + ω = (u 1 + u 13 ) dt + (u 1 v 1 + u 13 v 13 ) dxdt + (v 1 + v 13 ) dx. (4.35) It is found that h 11 = 1, h = 1, h 1 = h 1 = 0, hence H = 1 and K = 1. Solving the first two equations of (4.3) for u 3 and v 3, we obtain u 3 = v 3 = 1 u 1 v 13 u 13 v 1 [u 1 (v 1,t u 1,x ) + u 13 (v 13,t u 13,x )], 1 u 1 v 13 u 13 v 1 [v 13 (v 13,t u 13,x ) + v 1 (v 1,t u 1,x )]. (4.36) Using these results in the third equation of (4.3), the Gauss equation of H R,1 is obtained ( (v 1,t u 1,x )u 1 + (v 13,t u 13,x )u 13 ) x ( v 13(v 13,t u 13,x ) + v 1 (v 1,t u 1,x ) u 1 v 13 u 13 v 1 u 1 v 13 u 13 v 1 +u 1 v 13 u 13 v 1 = 0. (4.37) Let us summarize these results in the last Theorem of this section. Theorem 4.6. A nonlinear partial differential equation which admits an SO(, 1) Lax pair with u 1 v 13 u 13 v 1 0 is equation (4.37) or a subequation. Equation (4.37) is the 5 ) t

26 Gauss equation for H R,1, and u 1, u 13, v 1 and v 13 are given functions of ϕ and the partial derivatives of ϕ up to a certain order. Several examples of equations which are given by (4.37) after picking the u ij and v ij will be given to finish the Section. Example 1: Let u 13 = v 1 = 0, u 1 = cos(ϕ/), v 13 = sin(ϕ/), then (4.37) gives ϕ tt ϕ xx = sin(ϕ). Example : Taking u 13 = v 1 = 0, u 1 = cosh(ϕ/), v 13 = sinh(ϕ/), then (4.37) gives ϕ tt + ϕ xx = sinh(ϕ). Example 3: With u 13 = v 1 = 0, u 1 = v 13 = e ϕ. we have ϕ tt + ϕ xx = e ϕ. 5. MAURER-CARTAN COCYCLES AND PROLONGATIONS To introduce the idea of cocycle, some definitions and theorems will be given. This section will see some development and applications of the ideas in Section 3. There are several equivalent definitions of integrability of a set of 1-forms ω 1,, ω r defined on a manifold M of dimension n with r < n. Definition 5.1. A set of r linearly independent 1-forms ω 1,, ω r is called completely integrable if there exists in any neighborhood U M local coordinates x 1,, x n such that ω i = α i j dxj, i = 1,, r, (5.1) such that α i j are functions which are locally defined on U such that det(αi j ) 0. As a direct consequence of this definition, we have that the equations ω 1,, ω r define a local integral manifold of dimension n r. A necessary and sufficient condition for a set of 1-forms ω 1,, ω r to be completely integrable is provided by the Frobenius Theorem. Theorem 5.1. (Frobenius) A set of 1-forms ω 1,, ω r on a manifold M of dimension n (r < n) such that det(α i j) 0 is completely integrable if this set is closed dω i = ω j τ i j, (5.) where τj i, 1 i, j r are locally defined 1-forms. Let G be an n-dimensional connected Lie-group and let Ω M (G) be the exterior algebra of left-invariant forms, the Maurer-Cartan forms Ω MC (G) = Ω k MC (G) where (G) is the collection of left-invariant k-forms on G. Ω k MC 6

27 If we take as a basis of Ω 1 MC (G) the 1-forms ω1,, ω r, then every form ω on G is of the form ω = α i1 i p ω i1 ω ip, (5.3) i 1 < <i p where α i1 i r are C -functions on G and the p-form ω is left-invariant if and only if the functions α i1 i r are constant. Thus, Ω MC (G) is the exterior algebra over R generated by Ω 1 MC (G). By the Frobenius Theorem, Ω MC(G) is closed in the following sense. Definition 5.. Let Γ 1 be a vector space of finite dimension n on which an exterior differential operator d : Γ 1 Γ 1 Γ 1 is given. The exterior algebra Γ generated by Γ 1 and d will be called a Maurer-Cartan algebra. On Γ, we have that ddω = 0, d(ω + η) = dω + dη, d(ω η) = dω η + ( 1) k ω dη, where ω is a k-form. Therefore, Ω MC (G) is a Maurer-Cartan algebra and will be called the Maurer-Cartan algebra of the Lie group G. If ω i is a basis in Γ i, then we have dω i = c i jk ωj ω k, c i jk = ci kj. (5.4) The c i jk are called structure constants of the Maurer-Cartan algebra Γ with respect to this basis. Definition 5.3. Let Γ be a Maurer-Cartan algebra generated by a vector space Γ 1 of dimension n and M a connected manifold also of dimension n. Let Ω(M) be the set of exterior differential forms on M and Tm the cotangent space of M at m M. Then M is called a Maurer-Cartan space if there exists a morphism ϕ : Γ Ω(M), such that at every point m M, the mapping ϕ Γ 1 : Γ Tm is a bijection. It follows that all co-tangent spaces of a Maurer-Cartan space are isomorphic, and hence every connected open sub-manifold of a Maurer-Cartan space is again a Maurer- Cartan space. If we have a Maurer-Cartan basis {ω 1,, ω n } of Γ with structure constants c i jk, then in the Γ-space M we have, due to the morphism ϕ, n 1-forms τ 1,, τ n on M which form a basis of Tm at every point m of M such that dτi = c i jk τj τ k. The 1-forms τ i = ϕ(ω i ), i = 1,, n satisfy the same structural equations as those of the Maurer-Cartan algebra Γ. The idea of Γ-cocycles play an important role in the treatment of evolution equations. Definition 5.4. Let M be a manifold. A Γ-cocycle is a morphism χ : Γ Ω(M). This need not be an injection or surjection. A Γ cocycle means only that we have on M 1-forms σ 1,, σ n such that σ i = χ(ω i ) which satisfy dσ i = c i jk σj σ k. These 1-forms need not necessarily be independent. For example, the trivial Γ-cocycle is given by σ 1 = = σ n = 0, and every Γ space M has the injection ϕ : Γ Ω(M) as a Γ-cocycle. These cocycles can be referred to as representative cocycles on account of the following Theorem. 7